The Manga Guide to Calculus

112 Chapter 3 Let’s Integrate a Function!

Example:

Calculate:

10 21

4

0

1

xd

x+

()

∫

We rst substitute the variable so that y = 2x + 1, or

xgy

y

==

−

()

1

2

.

Since y = 2x + 1, if we take the derivative of both sides, we get

dy = 2dx. Then we get

dx dy=

1

2

.

Since we now integrate with respect to y, the new interval of integra-

tion is obtained from 0 = g(1) and 1 = g(3) to be 1 – 3.

*

10 21 10

1

2

531 242

4

0

1

4

1

3

4

1

3

55

xdxydy ydy+

()

===−=

∫∫∫

The Power Rule of Integration

In the example above we remembered that 5y

4

is the derivative of y

5

to finish

the problem. Since we know that if F(x) = x

n

, then F

′ 

(x) = f(x) = nx

(n + 1)

, we

should be able to find a general rule for fi ...